The nodal set of solutions to some elliptic problems: Singular nonlinearities

This paper deals with solutions to the equation-Delta u = lambda(+ )(u(+))(q-1) - lambda(-)(u(-))(q-1 )in B-1where lambda(+), lambda(-) > 0, q is an element of (0, 1), B-1 = B-1(0) is the unit ball in R-N, N >= 2, and u(+) := maxu, 0, u(-) := max-u, 0 are the positive and the negative part of u, respectively. We extend to this class of singular equations the results recently obtained in [25] for sublinear and discontinuous equations, 1 <= q < 2, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most N - 2 (locally finite when N = 2). As an intermediate step, we establish the regularity of a class of not necessarily minimal solutions.The proofs are based on a priori bounds, monotonicity formulae for a 2-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogeneous solutions. (C) 2019 Elsevier Masson SAS. All rights reserved.